Question: Simplify. Rewrite the expression in the form $a^n$. $\left(a^5\right)^{2}=$
Solution: $\begin{aligned} \left(a^5\right)^{2}&=a^{5\cdot 2} \\\\ &=a^{10} \end{aligned}$ This follows from the general rule $\left(x^m\right)^{n}=x^{m\cdot n}$. We can also see this is correct by expanding the powers. $\begin{aligned} \left(a^5\right)^{2}&=\underbrace{a^5\cdot a^5}_\text{2 times} \\\\\\ &=\underbrace{ \underbrace{a\cdot a\cdot a\cdot a\cdot a}_\text{5 times} \cdot \underbrace{a\cdot a\cdot a\cdot a\cdot a}_\text{5 times}} _\text{2 times} \\\\ &=a^{10} \end{aligned}$ In conclusion, $\left(a^5\right)^{2}=a^{10}$.